A function is graphed on a coordinate grid. As the domain values approach infinity, the range values approach infinity. As
the domain values approach negative infinity, the range values approach negative infinity. To which family of functions
might the described function belong?
absolute value
exponential
cube root
square root

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UpStudy AI · Accepted Answer

To determine which family of functions the described function might belong to, let's analyze the behavior of each option as the domain approaches positive and negative infinity:

1. **Absolute Value Function**: The absolute value function, $$ f(x) = |x| $$, approaches positive infinity as $$ x $$ approaches positive infinity, but it approaches 0 as $$ x $$ approaches negative infinity. Thus, it does not fit the description.

2. **Exponential Function**: An exponential function, such as $$ f(x) = e^x $$, approaches positive infinity as $$ x $$ approaches positive infinity, but approaches 0 as $$ x $$ approaches negative infinity. This also does not fit the description.

3. **Cube Root Function**: The cube root function, $$ f(x) = \sqrt[3]{x} $$, approaches positive infinity as $$ x $$ approaches positive infinity and negative infinity as $$ x $$ approaches negative infinity. This fits the description.

4. **Square Root Function**: The square root function, $$ f(x) = \sqrt{x} $$, approaches positive infinity as $$ x $$ approaches positive infinity, but is undefined for negative values of $$ x $$. Thus, it does not fit the description.

Based on this analysis, the function described belongs to the **cube root** family.
The function belongs to the cube root family.

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